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Derived equivalences of self-injective 2-Calabi--Yau tilted algebras

Anders S. Kortegaard

Abstract

Consider a $k$-linear Frobenius category $\mathscr{E}$ with a projective generator such that the corresponding stable category $\mathscr{C}$ is 2-Calabi--Yau, Hom-finite with split idempotents. Let $l,m\in\mathscr{C}$ be maximal rigid objects with self-injective endomorphism algebras. We will show that their endomorphism algebras $\mathscr{C}(l,l)$ and $\mathscr{C}(m,m)$ are derived equivalent. Furthermore we will give a description of the two-sided tilting complex which induces this derived equivalence.

Derived equivalences of self-injective 2-Calabi--Yau tilted algebras

Abstract

Consider a -linear Frobenius category with a projective generator such that the corresponding stable category is 2-Calabi--Yau, Hom-finite with split idempotents. Let be maximal rigid objects with self-injective endomorphism algebras. We will show that their endomorphism algebras and are derived equivalent. Furthermore we will give a description of the two-sided tilting complex which induces this derived equivalence.
Paper Structure (6 sections, 23 theorems, 49 equations, 3 figures)

This paper contains 6 sections, 23 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Derived equivalences
  4. Examples
  5. Cluster-tilting objects from C(D2n)
  6. Example from symmetric (k,n)-Postnikov diagrams

Key Result

Theorem A

Let $\mathcal{E}$ be $k$-linear Frobenius category with projective objects $\mathop{\mathrm{add}}\nolimits(r)$ for some $r\in \mathcal{E}$, such that the associated stable category $\mathcal{C} \coloneqq \underline{\mathcal{E}}$ is 2-CY and Hom-finite with split idempotents. Let $l,m\in \mathcal{C}$ making $\underline{A}$ and $\underline{B}$ derived equivalent. The subscript $\subseteq 1$ denotes

Figures (3)

  • Figure 1: Auslander-Reiten quiver of $\mathop{\mathrm{mod}}\nolimits(kD_{8})$.
  • Figure 2: To the left is a (3, 9)-Postnikov diagram, and to the right is its associated quiver, with the vertices on the outer boundary being the frozen vertices.
  • Figure 3: To the left is a (3, 9)-Postnikov diagram, and to the right is its associated quiver, with the vertices on the outer boundary being the frozen vertices.

Theorems & Definitions (44)

  • Definition
  • Definition 1.1
  • Theorem A: \ref{['cor:Ptilting']}
  • Corollary B: \ref{['cor:postnikov_example']}
  • Definition 2.2
  • Theorem 2.3: zhou2011maximal
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 34 more